exp_error_10_at_080454125
plain-language theorem explainer
This definition supplies the rational error term for the tenth-order Taylor remainder of exp at 0.80454125. Researchers certifying forced muon and tau masses in the Recognition Science lepton ladder cite it to close numerical bounds derived from T9 electron mass and cube geometry. The body evaluates the scaled remainder formula directly from the input fraction without external lemmas.
Claim. Define the error term $e_{10} := (80454125/100000000)^{10} · 11 / (10! · 10)$. This rational bounds the remainder after ten terms in the Taylor series for the exponential at the point 0.80454125.
background
The module establishes that the muon and tau masses are forced from the electron structural mass (T9) together with step functions built from E_passive = 11, cube faces = 6, wallpaper groups W = 17, the fine-structure constant alpha, and pi from spherical geometry. All inputs trace to the eight-tick octave and phi-ladder of Recognition Science. The point 0.80454125 appears inside the exponential that encodes one of these geometric steps when converting between the phi-ladder rungs and observed mass ratios.
proof idea
The definition is a direct algebraic expression. It binds the rational x = 80454125/100000000 and returns x^10 multiplied by 11 divided by (10! · 10). No lemmas or tactics are applied; the body is a pure computation of the standard Lagrange-form remainder scaled by the factor 11.
why it matters
The term is consumed by the lemmas exp_080454125_lower and exp_080454125_lower_q, which convert it into a strict lower bound on exp(0.80454125). Those bounds close the numerical gap between the T9 electron mass and the predicted muon/tau values, replacing the two axioms formerly present in LeptonGenerations.lean with theorems. It thereby completes the T10 necessity claim that the entire lepton ladder is determined by cube geometry and the golden ratio without additional hypotheses.
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